According to many of our Hanafi scholars, the Sunnah mu'kkadah after Zuhr, Maghrib, Isha, and Jumu'ah prayers should be fulfilled immediately after the prayer with nothing more than a brief pause and very few adhkaar. Some say that delaying the prayer more than that reduces the reward of the Sunnah, while I have heard others say that if the pause is too long, the Salah ceases to be the Mu'akkadah and just becomes nafl. On the other hand, some scholars say that there is no harm in performing the adhkaar before the Sunnah. But the former view seems to be the prevalent view today.
However, I've also seen a passage from Ibn Abidin which was quoted by Shaykh Faraz Rabbani that it is better to delay one's sunnah to pray it at home, provided that the interval is not unnecessarily long and that one does not miss the Sunnah. This seems to contradict the sense of "immediacy" that is often conveyed, for one may live further from the masjid and could perform adhkaar on the way home, etc.
Do we find any narrations from the Sahabah or Salaf that convey this sense of immediacy? And how do we reconcile this view with the view that it is better to pray the Sunan at home?
Imam al-Tahawi reports in his Sharh Ma‘ani al-Athar that, according to Imam Abu Hanifah, Imam Abu Yusuf and Imam Muhammad, it is fine to pray the sunnah mu’akkadah salahs in the masjid but it is better to pray them at home. (Sharh Ma‘ani al-Athar, 1:339)  Later Fuqaha have clarified, however, that this is only when one does not expect that if he were to return home he would become distracted and forget to pray the sunnah mu’akkadah salah altogether. In that case, he should perform them in the masjid. (al-Muhit al-Burhani, Idarat al-Qur’an, 7:503) 
In al-Asl, Imam Muhammad - in reporting the opinion of Imam Abu Hanifah, Imam Abu Yusuf and his own view - states that the imam of a congregational salah should not remain in his place after Zuhr, Maghrib and Isha, but should get up immediately. Moreover, he mentions that the imam may pray the sunnah mu’akkadah salah in the masjid if he chooses to do so but should move behind the congregation or somewhere besides where he led them in salah. The muqtadis may also pray in the masjid but it is better to move one or two steps before doing so. (al-Asl, 1:16-17)  The only reason for making this distinction between salahs that have sunnah mu’akkadah prayers after them and those that don’t would be because it is better to offer the sunnahs immediately after the fard without any significant delay.
Al-Zahidi reports in al-Qunyah that Zahir al-Din al-Marghinani felt that to delay the sunnah mu’akkadah till the ending time of the salah will cause it to cease being sunnah. However, ‘Ayn al-A’immah al-Karabisi said it will remain sunnah. (al-Qunyah, p. 40)  The latter view appears to be stronger, since al-Sadr al-Shahid (as well as al-Karabisi) said that to engage in conversation or anything else that goes against the nature of salah (like buying and selling) after the fard will not cause the sunnah prayer to cease being sunnah, but will only result in less reward, and he states that this is “the more correct” (asahh) view. (ibid.)  Ibn al-Humam and Ibn Amir Hajj also favour this view. (Halbat al-Mujalli, 2:226) They clarify that it is only recommended (awla/mustahabb) to not leave a gap between the fard and sunnah (Halbat al-Mujalli, 2:225-6), and hence to cause a delay is khilaf al-awla - which to do so is not blameworthy but only results in less reward.
But, as Ibn al-Humam and Ibn Amir Hajj also clarify, this only applies to the situation that the sunnah salahs are offered in the masjid. If a person is praying them at home, the time taken to reach the house will not cause any decrease in reward and will not be khilaf al-awla - as it is in fact better to pray them at home. (ibid. 2:226; Radd al-Muhtar, Dar ‘Alam al-Kutub, 2:247) 
وخالفهم في ذلك آخرون فقالوا: التطوع فى المساجد حسن غير أن التطوع فى المنازل أفضل منه....لقول رسول الله صلى الله عليه وسلم: خير صلاة المرء في بيته إلا المكتوبة؛ وهذا قول أبي حنيفة وأبي يوسف ومحمد بن الحسن رحمهم الله (شرح معاني الآثار، ج١ ص٣٣٩)
وفى الجامع الأصغر: إذا صلى المغرب فى المسجد بالجماعة يصلي ركعتي المغرب فى المسجد إن يخاف أنه لو رجع إلى بيته يشتغل، وإن لا يخاف فالأفضل أن يصلي في بيته لقوله صلى الله عليه وسلم: صلاة الرجل فى المنزل إلا المكتوبة (المحيط البرهاني، ج٧ ص٥٠٣)
قلت: أرأيت الإمام إذا فرغ من صلاته أيقعد في مكانه الذي يصلي فيه أو يقوم؟ قال: إذا كانت صلاة الظهر أو المغرب أو العشاء فإنني أكره له أن يقعد في مقعده حين يسلم وأحب إلي أن يقوم...فإن أراد فى الظهر والمغرب والعشاء أن يصلي تطوعا...يتأخر فيصلي خلف القوم أو حيث أحب من المسجد ما خلا مكانه الذي يصلي بهم فيه. قلت: فالذين خلفه أيصلون في أمكنتهم التي صلوا فيها أي يتنحون؟ قال: إن فعلوا فلا بأس، ويتنحون خطوة أو خطوتين أحب إلي. (الأصل، ج١ ص١٦-١٧)
ظم أخر السنة بعد الفرض ثم أداها في آخر الوقت لا يكون سنة عك يكون سنة (القنية ص٤٠)
عك الكلام بعد الفرض لا يسقط السنة لكن ينقص ثوابه؛ صح وكل عمل ينافي التحريمة أيضا، قال رض وهو الأصح (القنية، ص٤٠)
ثم قال: وأفاد شيخنا أن الكلام فيما إذا صلى السنة في محل الفرض لاتفاق كلمة المشايخ على أن الأفضل فى السنن حتى سنة المغرب المنزل؛ أي فلا يكره الفصل بمسافة الطريق (رد المحتار، ج٢ ص٢٤٧
This is a issue i investigated years ago. Frankly said from a point of evidence the Hanafi's have based their ruling on this hadith:
Aisha (may Allah be pleased with her) said:
"The Messenger of Allah (peace and blessings be upon him), after he would say his salaams [in the end of the prayer], wouldn't sit except for the time it took for him to say 'Allahumma anta al-salamu wa minka al-salamu tabaarakta ya dhal jalaali wal-ikram!'"
(Reported by Muslim Ahmad, Ibn Majah and al-Tirmidhi, with slight variations in wording)
But Shah Waliullah Dahlawi mentioned in his Hujjatul Baligha that this hadith means explained that this hadith means that the Prophet (salallahu alayhi was sallam) only sat for that duration towards the qiblah after the taslim, thereafter he (salallahu alayhi was sallam) turned to the congregation as is mentioned in other ahadith.
Hadith that after fard there should be delay to perform the post-sunnah prayer:
Narrated Al-Azraq ibn Qays: An imam of ours, whose kunyah (surname) was AbuRimthah, led us in prayer and said: I prayed this prayer, or one like it, with the Prophet (ﷺ). AbuBakr and Umar were standing in the front row on his right and there was a man who had been present at the first takbir in the prayer. The Prophet of Allah (ﷺ) offered the prayer, then gave the salutation to his right and his left so that we saw the whiteness of his cheeks, then turned away as AbuRimthah (meaning himself) had done. The man who has been present with him at the first takbir in the prayer then got up to pray another prayer, whereupon Umar leaped up and, seizing him by the shoulders, shook him and said: Sit down, for the People of the Book perished for no other reason than that there was no interval between their prayers. The Prophet (ﷺ) raised his eyes and said: Allah has made you say what is right, son of al-Khattab. Abu Dawud said: Sometimes the name of Abu Umayyah is narrated instead of Abu Rimthah.
(Abu Dawud 1007)
The evidence for this separation mentioned is the report narrated by Muslim in his Sahuh (1463) from Mu‘aawiyah (may Allah be pleased with him) who said: When you have prayed Jumu‘ah, do not follow it immediately with another prayer until you speak or leave (the mosque), because the Messenger of Allah (blessings and peace of Allah be upon him) instructed us to do that, not to follow one prayer immediately with another until we spoke or left (the mosque).
Some hadith clearly mention to do a specific dhikr after the fard.
One of the clearest ahadith in this regard is, “He will not be unsuccessful who says or does the following mu’aqqibat after the obligatory prayers: Thirty three times Subhanallah, thirty three times Alhamdulillah and thirty four times Allahuakbar.”
Sheikh Al-Albani (may Allah have Mercy on him) said, “Mu’aqqibat mean the words said after the prayers (salah). This hadith is evidence that this dhikr (Remembrance) should be said directly after the obligatory prayers.”
Halabi mentions in Saghiri a statement of Halwani. He says if is someone has a mutad wird that person should stand up from the place he offered his fard prayer and will read standing his wird. Or he will sit down in any side of the masjid and read his mutad wird, and that both practices are related from the sahaba (radiallahu anhum).
Ibn Abi Shaybah narrates with an authentic chain to the Basran Tabi'i, Abu Mijlaz (ca. 40 - 100 H), that he said: "Every Salah after which there is Sunnah, then turn away - besides Fajr and 'Asr." (Musannaf Ibn Abi Shaybah, no. 3105) The only reason for making a distinction between Salahs "after which there is Sunnah" and Salahs after which there is no Sunnah would be because it is recommended to pray the Sunnahs immediately after the Fard without any significant delay.
Ibn Abi Shaybah and 'Abd al-Razzaq narrate with their chains to 'Abdullah ibn Mas'ud (radiyallahu 'anhu) that after completing the Salah he would shift right away, either turning or getting up. (ibid. no. 3097, Musannaf 'Abd al-Razzaq, no. 3221) The Prophet's (sallallahu alayhi wasallam) practice of not sitting for long after the Fard can be interpreted as both turning to the side and getting up. It may be that in some Salahs he would get up and in other Salahs he would remain seated and turn to the left or right. The Hanafi understanding is that in Salahs after which there is Sunnah, the Prophet (sallallahu 'alayhi wasallam) would get up very quickly after the Fard; but in Salahs after which there is no Sunnah (i.e. Fajr and Asr), he would either remain seated and turn towards the congregation or leave.
Ibn Abi Shaybah narrates from Abu Razin that he prayed behind Ali (radiyallahu 'anhu) and he got up straight after leading them in Salah. (Musannaf Ibn Abi Shaybah,. no. 3099). He narrates from Abu 'Ubaydah ibn al-Jarrah (radiyallahu 'anhu) that when he would make salam, it was as if he was sat on heated stones until he got up; meaning that after the salam, he sat only for a very short while before getting up. (no. 3101) 'Abd al-Razzaq al-San'ani reports something similar about Abu Bakr (radiyallahu 'anhu) (Musannaf 'Abd al-Razzaq, no. 3214, 3215, 3231)
It is reported from the Sahabah, Ibn Abbas, Ibn al-Zubayr, Abu Sa'id and Ibn 'Umar (radiyallahu 'anhum), that they said one should not pray the Sunnah until he moves away from the place he prayed his Fard. (Musannaf ibn Abi Shaybah, no. 6067) It is also reported from the eminent Tabi'i, 'Amir al-Sha'bi, that one should not perform the Sunnah until he moves one or two steps away from the place where he prayed the Fard. (ibid. no. 6068) (One will notice that this is identical to what Imam Muhammad mentioned in the passage of al-Asl quoted from him earlier).
The hadith of Abu Rimthah in Sunan Abi Dawud (which contains some weakness) discouraging prayer immediately after the Fard Salah is understood to be referring to praying the Sunnah in the same spot straight after the Fard - which is not encouraged according to the Hanafis. It is better to move away from the place where the Fard was performed. (Ghunyat al-Mutamalli, p. 342) Simply moving away a couple of steps would be sufficient to create the short gap recommended in the hadiths of Abu Rimthah and Mu'awiya (radiyallahu 'anhuma).
The authentic transmission from Shams al-A'immah al-Halwani is as follows: "There is no harm in reciting the awrad between the Fard and Sunnah." This is not in disagreement with what was mentioned earlier. While there is no harm in doing this, it is better (awla/mustahabb) to delay the adhkar and awrad until after the Sunnah.
From Abdullaah bin Rabaha from a man who was from the Companions of the Prophet -sallAllaahu alayhi wa sallam- that the Messenger of Allaah had prayed the Asr prayer, and this man (immediately) stood up to pray. `Umar saw him and said to him sit down, since indeed the people of the book were destroyed because they did not leave a gap after their (obligatory) prayer.
The Prophet -sallallaahu `alayhi wa sallam- said to him:
أحسن ابن الخطاب
'You have done well O Ibn Khattab!'
[Ref: Collected by Ahmad, its Sanad is Saheeh as declared by al Haythami in Majmu' az-Zawa'id 2/237 and in Silsilah al-Saheeha 2549 or 6/105.]
Shaykh Albaani said: 'This hadeeth is a clear text in that it is prohibited to hasten to pray the Sunnah prayer after the obligatory prayer without speaking or leaving the Masjid, like many of the non-Arabs do, and especially the Turks. Indeed we see them in the Haramain Sharafain (the holy Masjid in Makkah and the Prophet's Masjid in Madina), the Imaam is about to give the Salam from the prayer, except that these people hasten, from over here and over there standing to pray the Sunnah! [Ref: 'Silsilah Saheehah' No. 2549 6/105]
I found the following in addition for the Hanafi stance:
It is reported that the Prophet (peace be upon him) would immediately stand up after his prayers. [Musannaf Abdur Razzaq 2/246]
It is reported That Abu Bakr As-Sideeq after he completed the prayers (as the Imam) he would immediately stand up. The Narrator states "He would do so, so quickly as though he were on burning coal" [Musannaf Abdur Razzaq 2/242-246]
It is reported that Abdullah Ibn Masud said:"When the imam completes the Tasleem, he should immediately stand up, and if not then he should at least turn away. He was asked as to whether it would suffice for the imam to move from his spot and face the Qiblah? He replied; "He should face the east or west (i.e. He should face a direction other than the Qiblah." [Musannaf Abdur Razzaq 2/242-243]
I saw some similar reports in Shatibi's al Itisam, even one report that Ibn Umar (ra) said it is bidah to remain seated for the imaam after the taslim.